\documentclass{minimal}
\begin{document}

\begin{proof}
  This is the proof of Theorem 1.
  \begin{equation}
    1+1=2
  \end{equation}
\end{proof}

Hello

\begin{equation}
  1+1=2
\end{equation}

\begin{equation*}
  1+1=2
\end{equation*}

World

\begin{equation}
  f(x) = 1 + e^x
\end{equation}

\end{document}
